This paper presents stable formulation of fundamental alternating-direction-implicit finite-difference time-domain (FADI-FDTD) method for multiterm, doubly, second-order dispersive media. The formulation is achieved by decomposing the electric and magnetic susceptibility functions into first-order poles and it is derived based on currents constitutive relations. It is conveniently applicable for most commonly used second-order dispersive models, such as Lorentz and Drude models, and equally applicable to first-order Debye model. The extension for full 3-D doubly dispersive media using FADI-FDTD method makes the resultant update equations much more concise and simpler than using conventional ADI-FDTD method. The number of field variables and update coefficients are greatly reduced at the right-hand sides, with only first-order spatial differencing and no dispersive terms and magnetic update coefficients in the implicit E update equations. All these contribute to conciseness and programming simplicity, as well as leading to higher efficiency due to much less memory indexing overhead and fewer arithmetic operations. Most importantly, our present formulation is stable, while the contemporary second-order differential equation formulation has potential instability as to be demonstrated. The stability analysis is performed using Fourier method by examining the eigenvalues of the Fourier amplification matrix numerically.